1. Field of the Invention
The present invention relates to a system for and a method of analyzing an electrocardiogram (ECG), and more particularly to a system for and a method of analyzing an ECG, capable of analyzing an ECG signal generated by the pulsation of the heart in a chaotic fashion using an attractor reconstruction which is one of qualitative analysis methods and using a correlation dimension which is one of quantitative analysis methods.
2. Description of the Prior Art
ECG signals generated by the pulsation of the heart can be interpreted by the chaos theory. Chaos systems which are analyzing systems utilizing the chaos theory belong to the category of nonlinear deterministic dynamic systems. Here, the "dynamic system" means the system wherein its system controlling state varies by the lapse of time. The "deterministic system" means the system wherein its system state after a given time elapses can be accurately determined in so far as a state variable controlling the dynamic system and a current initial condition are known. Such nonlinear deterministic dynamic systems are divided into four classes, namely, the equilibrium point system, the periodic of limit cycle system, the quasi-periodic or torus system, and the chaos system. As compared to other systems, the chaos system disables a prediction of the future because it is very sensitive to the initial condition, in spite of the deterministic presence thereof. The chaos system has important features, that is, an infinite cycle thereof (no presence of any cycle) and a state variation very sensitive to the initial condition. Methods for analyzing and establishing chaos phenomena occurring in nature are classified into the qualitative analysis method based on the topology of attractor and the quantitative analysis method based on the dimension. Such a classification is based on the fact that a specified chaos system has an intrinsic attractor topology and an intrinsic dimension.
The qualitative analysis method is to analyze a specified system using a path along which the trajectory of a state variation moves by the lapse of time and the topology of the whole trajectory. As described above, chaos systems are nonlinear deterministic dynamic systems. Information essentially required in a dynamic system to indicate how the system varies by the lapse of time is referred to as "state". The equation expressing the state of a chaos system, that is, the state equation has the form of a differential equation or the form of a difference equation. A variation in state occurring after a given time lapses under a condition that an initial condition has been given is referred to as the solution of a state equation or the trajectory on a state plane. The point on which the trajectory converges. "Attractor construction" is to derive the solution of the state equation of a specified chaos system and to indicate the derived solution in a state space. In other words, the attractor may be referred to as a graphic method for indicating acts and characteristics of a specified system in a state space. Where an attractor construction is carried out in a two-dimensional state space, a value X of the state occurring after a given time lapses is indicated on an X-axis of the state space. At the same time, a differential value X' of the state value indicated on the X-axis is indicated on an Y-axis of the state space. It is impossible to find all state variables and parameters of a specified chaos system existing in the nature and derive a state equation of the system. Takens verified that where a variation in one of the state variables importantly affecting the system is experimentally measured in the form of time series data and a reconstruction of attractor is carried out based on the measured time series data, the reconstructed attractor has a similar topology to the original attractor. The reconstruction of attractor is achieved using sampled time series data, embedding dimension and delay time.
Where m-dimensional vectors are constructed from time series data {X.sub.1, X.sub.2, X.sub.3, . . . , X.sub.N } (here, N is a sufficiently large value) detected for an attractor reconstruction, they are expressed as follows: EQU Y.sub.i ={X.sub.i, X.sub.i+p, X.sub.i+2p, X.sub.i+3p, . . . , X.sub.i+(m-1)p }
where, i=0, 1, 2, 3, . . . , N PA1 S.sub.0 ={X.sub.4, X.sub.1 } PA1 S.sub.0 ={X.sub.5, X.sub.2 } PA1 S.sub.0 ={X.sub.6, X.sub.3 } PA1 S.sub.0 ={X.sub.7, X.sub.4 } . . . PA1 S.sub.N-4 ={X.sub.N, X.sub.N-3 } PA1 Y.sub.1 ={X.sub.1, X.sub.1+p, X.sub.1+2p, X.sub.1+3p, . . . , X.sub.1+(m-1)p } PA1 Y.sub.2 ={X.sub.2, X.sub.2+p, X.sub.2+2p, X.sub.2+3p, . . . , X.sub.2+(m-1)p } PA1 Y.sub.3 ={X.sub.3, X.sub.3+p, X.sub.3+2p, X.sub.3+3p, . . . , X.sub.3+(m-1)p } . . . PA1 Y.sub.i ={X.sub.i, X.sub.i+p, X.sub.i+2p, X.sub.i+3p, . . . , X.sub.i+(m-1)p } . . . PA1 Y.sub.j ={X.sub.j, X.sub.j+p, X.sub.j+2p, X.sub.j+3p, . . . , X.sub.j+(m-1)p } . . . PA1 Y.sub.M ={X.sub.M, X.sub.M+p, X.sub.M+2p, X.sub.M+3p, . . . , X.sub.M+(m-1)p } PA1 where, m represents the embedding dimension, p represents the delay time, and M represents the number of vectors identical to the number of data N.
Using the embedding dimension m and the delay time p, the reconstructed time series data are derived as follows: ##EQU1## where, t represents the time of the reconstructed time series data.
The reconstruction of time series data is carried out in a two-dimensional plane when the embedding dimension m is 2 and in a three-dimensional space when the embedding dimension m is 3. However, time series data are reconstructed generally in two-dimensional planes. For example, where the embedding dimension m and the delay time p are 2 and 3, respectively, the following vectors are reconstructed from time series data {X.sub.1, X.sub.2, X.sub.3, . . . , X.sub.N } measured:
Each of the reconstructed vectors has two vector components one being associated with an X-axis while the other being associated with a Y-axis. As S.sub.0, S.sub.1, S.sub.2, . . . , S.sub.N-3 are sequentially plotted in a two-dimensional state space, reconstruction of the attractor of the specified chaos system is achieved. In other words, the plotted results indicate the reconstructed attractor. FIG. 1 illustrates the case wherein vectors S.sub.0 and S.sub.1 are plotted in a two-dimensional state space. Such an attractor reconstruction can be meaningful only when the delay time for the attractor reconstruction is appropriate. If the delay time is excessively small, the reconstructed attractor has a topology shaped into a diagonal line because of X(0).congruent.X(P).congruent.X(2P). In the case of an excessively large delay time, the association between X(0) and X(P) disappears due to noise. As a result, the attractor reconstruction is meaningless. It, therefore, is very important to select an appropriate delay time. Generally, such a delay time is known to be a quarter of the basic cycle of a dynamic system and may be derived using an autocorrelation function. FIGS. 2A to 2C illustrate a reconstructed attractor, time series data and a frequency characteristic of an ECG chaos system, respectively.
The correlation dimension which is one of quantitative analysis methods for chaos systems will now be described. The reason why the correlation dimension is important in analyzing chaos systems is because a specified chaos system has an intrinsic correlation dimension value that we can not explain theoretically, but easily derive from experimental data based on the embedding dimension and delay time used to reconstruct a state space. For deriving such a correlation dimension value, m-dimensional vectors are first constructed from time series data as follows:
If two successive ones of the vectors derived as above have a distance .vertline.Y.sub.i -Y.sub.j .vertline. less than a given, very small value .epsilon. therebetween, they are regarded as having a correlation.
Correlation sum or correlation integral C(.epsilon.) is expressed as follows: ##EQU2##
Assuming that ".epsilon.-.vertline.Y.sub.i -Y.sub.j .vertline." in this equation is x, T(x) is 1 under a condition of x&gt;0 (T(x)=1) and 0 under a condition of x&lt;0 (T(x)=0). Accordingly, the correlation sum C(.epsilon.) is a step function serving as a counter.
The correlation dimension D.sub.c is defined as follows: ##EQU3##
When .epsilon. is excessively large, all vectors are correlated together. In this case, the correlation sum C(.epsilon.) becomes 1, thereby causing the correlation dimension D.sub.c to be zero (D.sub.c =0). In the case of C(.epsilon.).varies..epsilon..sup.v, the correlation dimension D.sub.c corresponds to V with respect to .epsilon. which is variable. Since .epsilon..sup.v is effective only in the range of .epsilon. in which the correlation sum C(.epsilon.) has a linearity, it is very important to calculate the correlation dimension by finding a linear domain defined between two optional points on a log C(.epsilon.)-to-log .epsilon. plot curve. However, there is a difficulty to experimentally calculate the correlation dimension. In other words, there is a possibility of experimental error because the linearity between two points on the log-log plot curve is indefinitely exhibited due to the presence of more or less noise signals in experimental data. It, therefore, is necessary to determine whether experimental data from a specified chaos system is a chaos signal or a meaningless noise signal. This determination can be accomplished using a correlation dimension graph for the chaos system. FIGS. 3A and 3B show a method for determining whether experimental data from a specified chaos system is a chaos signal or a meaningless noise signal, respectively. FIG. 3A shows an attractor reconstructed using an appropriate delay time. FIG. 3B shows a graph drawn by plotting a variation in the correlation dimension value of the reconstructed attractor shown in FIG. 3A upon varying the embedding dimension into .alpha., .beta., .GAMMA. . . . , in this order. Where correlation dimension values plotted converge algebraically, their convergence value becomes the correlation dimension of the system. In this case, the experimental data is the chaos signal. On the contrary, where the correlation dimension values diverge, the experimental data is the noise signal. FIGS. 4A to 4C are diagrams respectively showing methods for deriving correlation dimension values at different embedding dimensions of .alpha., .beta. and .GAMMA.. Deriving an embedding dimension is the same as deriving the slope of the linearity-exhibited portion of an interval in which the value of log C(.epsilon.) varies. The correlation dimension has not integral dimensions, but has fractal dimensions. This feature of the correlation dimension is one of the important features of the chaos.
However, such a chaos theory has been hardly utilized to analyze ECG signals generated by the pulsation of the heart, in spite of the fact that ECG signals have chaotic features, namely, unique attractor topologies and correlation dimension values individually depending on the health and psychological conditions.